Article here. Essentially, tamarin monkeys showed a capacity for recognizing a pattern of phonemes, and then recognizing when a novel pattern appeared (if an affix was used in the wrong place).
Primates can frequently recognize language-like stimuli when exposed to them, but whether they can be trained to generate them is another question. (And we know they don't generate them spontaneously.) It seems to me there are two take-homes here. First, that many primates (including those not even that closely-related to us) have the hardware for linguistic pattern recognition. This raises the question of whether other linguistic substrates delivered over a time sequence could evince similar responses: chains of images for example. It would be interesting to compare human and non-human primates in these experiments (with sound and non-sound grammar).
The second, and to me more interesting, point to investigate is to what degree non-human primates can associate the patterned-sounds they're hearing with semantic content. No, it wasn't a concern at all in this experiment, but it's crucial to the development of language. While Endress et all conducted this experiment with particular reference to grammar acquisition, a model of vocabulary acquisition in human children uses much the same pattern-recognition skill: children learn words by looking for sequence rules, and take notice when they're violated. A chid learning English for the first time has no way to know that "word boundary" is two words, and where the dividing line is, until they figure out that English words don't often start with "db"; in fact words don't even have that sound combination so much (and of course they hear the words separately). The tamarins were doing some form of this.
What the tamarins can't do that childre can - or at least no one's shown that tamarins can do it, and I don't think anyone expects it - is that once they've parsed out the elements and learned the order they usually occur in, they can build a network and assign each element to an object or attribute they see in the real world. I wouldn't be blown away if a tamarin learns to be "surprised" (as in this experiment) by an "o" coming at the beginning of a word, as opposed to at the end. What the tamarins won't ever learn is that the -o ending means that the word in question is an object that is receiving action, as opposed to performing it. Yet somehow every normal Japanese child learns exactly this by age 5, and lots of other such content-sound relations as well.
Animals clearly can associate a few sets of sounds with concrete, specific content. Anyone who has ever had to spell out "W-A-L-K" to a fellow human in front of a dog knows this. But the extent of language perception in non-human animals is an interesting question because it gives us a chance to do some comparative biology with reference to the following questions. How how many of these sets of sounds can the dog (or the tamarin) learn? How generalizable is the ability - i.e., given the internal states of the animal as they reflect the outside world, how broad or distinct are the categories that can be covered by a single signifier - your dog understands tree, but does it understand redwood? Plant? How complex a relationship between signifiers can the animal construct, e.g., can the animal tell a difference between walk, walked, and don't walk? And to what degree are these things related - that is, do human children get a mnemonic benefit from pinning the sounds they learn into a richer network of semantic content? Pinning down these differences to the activity of physical structures in our brains will go a long way to understanding how we acquire and process language.